elementarily equivalent

Conventions

All structures share a common signature; the first-order language $\mathcal{L}$ is the language determined by that signature.

Definition

The theory of a structure $\mathcal{M}\text{, }\operatorname{Th}(\mathcal{M})\text{,}$ is the set of all sentences of $\mathcal{L}$ that are true in $\mathcal{M}.$

Definition

Structures $\mathcal{M}$ and $\mathcal{N}$ are elementarily equivalent, (in symbols: $\mathcal{M}\equiv\mathcal{N})$ if and only if $\operatorname{Th}(\mathcal{M})=\operatorname{Th}(\mathcal{N})$.

Title elementarily equivalent ElementarilyEquivalent 2013-03-22 13:00:26 2013-03-22 13:00:26 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 03C99 theory